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Affine Kac–Moody Lie superalgebras of type $\widehat{B(0,n)}$

Affine Kac–Moody Lie superalgebras of type $\widehat{B(0,n)}$ behave very similar to the Affine Kac-Moody Lie algebra of type $A_{2n}^{(2)}$. For example, it was noted in Kac and Wakimoto - Integrable ...
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Relation between two Harish-Chandra homomorphisms

Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
S. D. Z's user avatar
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2 votes
0 answers
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graded reps of Lie algebras literature

I am currently studying 'advanced' representation theory from a physicist's perspective, including topics like super-Lie algebras. I've come across various gradings (excluding the ℤ2 grading), such as ...
iron's user avatar
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18 votes
3 answers
2k views

Mathematical motivation for supergeometry

Motivated by SUSY, mathematicians began to study $\mathbb{Z}_2$-graded mathematics, or super mathematics. In particular, one can formulate supergeometry just following Grothendieck style (even) ...
Estwald's user avatar
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4 votes
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smooth super scheme which is not smooth

I am following the very nice "Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes" by Bruzzo, Ruiperez and Polishchuk. I am having some problem in order to give ...
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symmetrization and invariants

Let $V$ be a supervector space over $\mathbb C$ and let $T^n(V):=V \otimes V \otimes \cdots \otimes V$ and let $S^n(V)$ be the super vectorspace of symmetric tensors. Then we have a cannonical ...
jack's user avatar
  • 599
4 votes
1 answer
122 views

Real forms of the general linear Lie superalgebra

I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}_{m|m}(\mathbb{C})$. The real forms of the simple complex Lie superalgebras were classified by ...
Alistair Savage's user avatar
1 vote
0 answers
35 views

Do the Tits-Kantor-Koecher construction extend to non unital Jordan algebras?

I have an infinite dimensional Jordan algebra which is not-unital. I would like to study a Lie algebra related to it (if it exists). Natural choice would be to look at the KKT construction, but ...
Dac0's user avatar
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9 votes
1 answer
309 views

What is the Lie superalgebra generated by permutations?

Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by $$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$ ...
WunderNatur's user avatar
1 vote
1 answer
245 views

Orthosymplectic superalgebra

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$ which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
jack's user avatar
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Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
double-function's user avatar
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Invariants of Lie superalgebras

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$ which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
jack's user avatar
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0 answers
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Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection

Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...
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What does “$d$-generated algebra” mean? [closed]

I would like to know what does ”$d$-generated algebra” mean? In particular “two-generated algebra”? @KonstantinosKanakoglou @LSpice Here is an example of a paper concerning “one-generated algebras”: ...
Amine Benayadi's user avatar
7 votes
1 answer
296 views

Semisimple super Lie algebras

There is a classification of simple Lie algebras in $\text{Vec}_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones. Question: How about ...
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6 votes
2 answers
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Lie powers of a graded vector space and Klyachko's theorem

Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...
Igor Khavkine's user avatar
7 votes
0 answers
107 views

Reference request: superconformal algebras and representations

I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...
winawer's user avatar
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3 votes
1 answer
244 views

Supersymmetry charge $Q$ as anti-linear and anti-unitary operator

We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$: $$ (-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0 $$ which defines the anti-...
wonderich's user avatar
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2 votes
1 answer
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Notation on supergeometry — parity

I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...
Eggon Viana's user avatar
9 votes
2 answers
298 views

Schur Weyl duality for the supergroup $\text{GL}(m|n)$

Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$. For every natural number $d$ we have a natural map $$\Phi_d:\mathbb{C} S_d\to \text{End}_G(V^{\...
Ehud Meir's user avatar
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1 vote
0 answers
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is this the correct universal property of free Lie superalgebras?

Consider a $\Bbb Z_2$ graded set $A$. Universal property of free Lie superalgebra $FLS(A)$: Let $\mathfrak g$ be a Lie superalgebra and let $\Phi: A \to \mathfrak g$ be a set map which preserves the $\...
GA316's user avatar
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1 vote
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Poisson reduction in odd/graded Poisson geometry?

I would like to know whether there is any literature on Poisson reduction of $\mathbb Z$- or $\mathbb Z_2$-graded Poisson algebras. A $\mathbb Z$-graded Poisson algebra with degree $p\in\mathbb Z$ ...
AlexArvanitakis's user avatar
1 vote
2 answers
376 views

Doubt in the Serre relation and the odd/even roots of a Lie superalgebra

Let $I = \{1,2,\dots,n\}$ and $S \subset I$. The set $I$ will be indexing the simple roots and $S$ will be indexing the odd generators of a Lie superalgebra. A real matrix $A=(a_{ij})_{i,j\in I}$ is ...
GA316's user avatar
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2 votes
1 answer
86 views

Sufficient conditions for unitarity of a representation of a Lie Superalgebra

Suppose we have a Lie superalgebra with triangular decomposition: \begin{equation} \mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-} \end{equation} I've seen it stated ...
dz16's user avatar
  • 61
1 vote
0 answers
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Quadratic Lie superalgebras

Let $(L,.)$ be a Lie superalgebra endowed with an even supersymmetric non-degenerate and invariant bilinear form $B$ (i.e $(L,.,B)$ is a quadratic Lie superalgebra). If we have the equality $B(x,y.z)=(...
Amine Benayadi's user avatar
7 votes
0 answers
121 views

Infinitesimal description of homogeneous supermanifolds

Lie's third theorem says that if $\mathfrak{g}$ is a real, finite-dimensional Lie algebra, then there is a unique (up to isomorphism) simply-connected Lie group $G$ whose tangent Lie algebra is ...
José Figueroa-O'Farrill's user avatar
10 votes
3 answers
1k views

Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
Nadia SUSY's user avatar
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Cartan integers for Lie superalgebras

Let $\mathfrak g$ be a basic classical simple Lie superalgebra (BCSLSA in short) with Cartan matrix $A$ of some simple system $\Pi$. Then $A$ satisfies the conditions that 1) $\frac{2a_{ij}}{a_{ii}} \...
GA316's user avatar
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2 votes
0 answers
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Super characters in the theory of Lie superalgebras

Weyl character formula for the finite dimensional complex semisimple Lie algebras plays a crucial role in the theory of highest weight modules, where for the highest weight module $V(\lambda)$ we ...
GA316's user avatar
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2 votes
1 answer
325 views

Category of typical representations for Lie superalgebras

I am interested in studying the category of typical representations over basic classical simple Lie superalgebras. In particular, I want to know 1) is this category semisimple (character determines ...
GA316's user avatar
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2 votes
0 answers
55 views

Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra. Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...
GA316's user avatar
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2 votes
0 answers
70 views

Dynkin diagram of Basic classical simple Lie superalgebras

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical simple Lie superalgebra with the root system $\Delta = \Delta_0 \cup \Delta_1$ and Dynkin diagram $\Gamma$. It is well-known ...
GA316's user avatar
  • 1,229
4 votes
1 answer
363 views

Typical and atypical modules for Lie superalgebras

There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only ...
GA316's user avatar
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4 votes
2 answers
391 views

Lie super algebra presentation of the Kähler identities

For any Kähler manifold $(M,h)$, with Lefschetz operators $L$ and $\Lambda$, and counting operator $H$, we have the following the well-known Kähler-Hodge identities: \begin{align*} [\partial,L] = 0, ...
Pierre Dubois's user avatar
1 vote
0 answers
79 views

GKO construction for (Super-)Virasoro algebras

I am reading the paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. In many places, the authors claim results without any justification, or with ...
soap's user avatar
  • 359
2 votes
1 answer
142 views

super Lyndon words

Lyndon words form a basis for Free Lie algebras. In this direction, I need the reference for the super Lyndon words for free Lie superalgebras. Given the definition of super Lyndon words, how to ...
GA316's user avatar
  • 1,229
2 votes
1 answer
129 views

Lyndon basis of free Lie superalgebras

Lyndon basis for Free Lie algebras is well known in the literature. My question is, what is the analogous combinatorial model for the case of free Lie superalgebras? what is the super analogous of ...
GA316's user avatar
  • 1,229
4 votes
0 answers
71 views

Reference Request: Representation Theory of Real Lie Superalgebras

Are there some references for the representation theory real lie superalgebras, specifically of $psu(1,1|2)$, and $u(1,1|2)$?
dz16's user avatar
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5 votes
2 answers
610 views

Serre relations for Lie Superalgebras

Finite dimensional complex simple Lie algebras are classified using Cartan matrices. One of the main ingredients is Serre Relations. Lets call this Cartan-Killing theory. I have the following ...
GA316's user avatar
  • 1,229
2 votes
0 answers
130 views

Coordinate ring of an equivariant embedding of a homogeneous projective variety

Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...
NongAm's user avatar
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10 votes
1 answer
563 views

Kazhdan-Lusztig equivalence for Lie super-algebras

Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which ...
Alexander Braverman's user avatar
2 votes
0 answers
73 views

odd positive roots of basic classical Lie superalgebras

In the Lie algebra case, positive roots are "almost ($S_{\alpha}$ permutes $\Delta^+ -\{\alpha\}$)" invariant under simple reflections. A similar statement I want to understand for Lie superalgebras. ...
GA316's user avatar
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2 votes
0 answers
226 views

action of Weyl group element on Weyl vector

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...
GA316's user avatar
  • 1,229
1 vote
1 answer
184 views

finite dimensional modules are highest weight modules [closed]

Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector. My feeling is, since $e_i$'s ...
GA316's user avatar
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1 vote
0 answers
75 views

how the borel subalgebras of p(n) look like except the standard one?

I am reading the book Musson: Lie superalgebras. In Chapter 3, on page 62, Lemma 3.6.8 tells about support of Borel subalgebra. I am confused about this. I am trying to do the proof of the Proposition ...
Shushma Rani's user avatar
7 votes
2 answers
794 views

Character formula for Lie superalgebras

The Weyl character formula and the denominator identity play important roles in the representation theory of classical simple Lie algebras and Kac-Moody Lie algebras over $\mathbb{C}$ Can you suggest ...
GA316's user avatar
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2 votes
1 answer
147 views

$P(1)$ strange type classical Lie superalgebras

I am reading the book "Lie superalgebras and enveloping algebras" by Ian M. Musson. The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $...
GA316's user avatar
  • 1,229
0 votes
1 answer
160 views

left ideals in Lie super algebras

Let $\mathfrak g$ be a Lie superalgebra. If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same? For me ...
GA316's user avatar
  • 1,229
2 votes
3 answers
521 views

Graph of a Lie super algebra

Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...
GA316's user avatar
  • 1,229
2 votes
0 answers
143 views

Two definitions of super-Virasoro algebra

Let $A=\mathbb C[x,\epsilon]$ where $x$ is an even variable and $\epsilon$ is an odd variable (thus $A$ is a commutative super-algebra). Let $\mathfrak g$ denote the Lie super-algebra of vector fields ...
Alexander Braverman's user avatar